Thin shell equations for circular pipe bends

Nuclear Engineering and Design 65 (1981) 77-89 North-Holland Publishing Company

77

THIN SHELL EQUATIONS FOR CIRCULAR PIPE BENDS J.F. WHATHAM Australian Atomic Energy' Commission, Research Establishment, Lucas Heights, N.S.W. Australia 2232 Received 2 December 1980

The equations for Novozhilov's linear first order thin shell theory are applied to the deformation of circular pipe bends, and the governing equations are presented for bends subjected to internal pressure and in-plane and out-of-plane forces and moments. Neglecting end effects, numerical results are given for flexibility and deflection factors for pure in-plane bending, deflection factors for pressure and displacement factors for out-of-plane bending under constant shear and torsion. Wall thicknesses up to 0.3 of the pipe radius are considered.

1. Introduction

The primary cooling circuit of the DIDO type research reactor at the Lucas Heights Research Establishment consists in the main of short runs of piping connected with flange ended elbows thus minimizing the heavy water inventory. As the stresses in the pipework from, for example, assembly misalignments or thermal expansion during start-up, depend on the flexibility of the elbows and little information has been published, a solution was sought for pipe bends with rigidly constrained ends subjected to various end loadings. Such solutions would also be applicable to the primary cooling circuits in power reactors where pipe bends occur adjacent to pressure vessel nozzles which have the effect of a rigid flange. The results of earlier analyses of pipe bends by the thin shell theory of Novozhilov [1] have been published elsewhere (Whatham [2]; Whatham and Thompson [3]; however, the procedure was only given in contracted form. The aim of this paper is to give the equations and their derivation for the pressurizing of pipe bends and, by superposition, for any combination of forces and moments acting on the ends, neglecting end effects for the present. The method differs slightly from that used in the above papers in that the forces, moments and displacements are obtained directly from the system equations instead of using Novozhilov's pseudo-displacement functions and deformation parameters. As in the previous work, let the thin shell assumptions be that: (i) the pipe wall is thin, (ii) normal stresses through the wall are negligible, and (iii) normals through the wall remain normal to it and unchanged in length.

2. Analysis

The pipe bend is represented by its middle surface, that is, an imaginary surface mid-way between the inner and outer surfaces, which has a certain stiffness. An element of the middle surface shown in fig. 1 is supposed to have forces To, Tn, Ton, Tno, N o, N, and moments M o, Mn, Mno, Mo, per unit length acting on 0029-5493/81/0000-0000/$02.50 © North-Holland

78

J. 1~: Whatharn / Thin shell equations fi)r ,:trcular pipe bend~

i

\,

i

7

~,

'

\~.

' \

' /"

s

<~c" .

<

l

//

\ Fig. 2. Pipe middle surface.

Fig. 1. Elementof pipe middle surface.

its edges and may have a force per unit area acting on it with components qo, q,, q,, in the coordinate directions. Fig. 2 shows the middle surface of a curved pipe in which displacements are denoted by u, t', w and a rotation by ~p. Coordinate 0 measures the angular position around the pipe cross-section relative to the extrados, and coordinate ,/locates the pipe cross-section from the pipe end, measured along the centre line. The latter was preferred to ¢ because it also applied to straight pipes. Coordinate T/= R e p / r .

(t)

At any pipe middle surface cross-section, such as that shown in fig. 3, there are assumed to be forces P, Q, T~ and moment M, acting per unit length of circumference. Comparing fig. 3 with fig. 1, Too, N o and M,o

Q

/ Fig. 3. Pipe middle surface boundary.

J.F. Whatham / Thin shell equationsfor circularpipe bends

79

have been replaced by P, Q giving a statiscally equivalent system if: 1 OMoo Q = N n + r O~0--

P = T,o + l Mno,

(2)

There are now eight unknowns which have to be determined for any pipe bend analysis and these are:

u, v, w, +,Mn, Tn, P, Q. Solving for these unknowns, the stresses in the pipe wall are:

transverse stress Ooe = T o / t + 12zMo/t 3, longitudinal stress %n = T n + 12zM./t3, t

shear stress S ( lgH °n° = °°n = -7 + z t3

S rt

ScosO) aRt '

(3)

where

vet( ou 02w) 002 '

TO= vT~ + 7Et ( gO Ou + w ') S = ~

P+6(l+v)

y = (t/r)2,

34° = vMn +---~- 00

r 00

00

'

H-2(3+3,)

P

2(1+v)

r 00

~0

E = Young's modulus,

u = Poisson's ratio,

8 = 1 + ( r / R ) cos 0.

For the sake of completeness the remaining forces and moments in fig. 1 are given by

Ton = S + ( H cos O/6R ),

Trio = S + H / r ,

Mo,~ = Mno = H, 10M o

sin 0

No - ~ ~0 + T h -

Nn=Q-r

1 OH ( Mn - M~ ) + ar a-~ '

(4)

1 OH O0

3. System equations

It is convenient to make the forces, moments and pressures non-dimensional as follows: To*, T~*, P*, Q* = (TO, Tn, p, Q ) / E t ,

M~, M,~ = (Mo, Mn)/rEt ,

q~, q,~, q* = (qo, qn, q , ) r / E t ,

(5)

J.F. Whatham / Thin shell equationsjbr circular pipe bends

80

and to regard the forces and moments as consisting of two components: T~ = Toa-~ Tob,

T~ - T~a + Tnb ,

M~

M~ = Mn, + Mnb,

P* = Pa + Pb,

Q* = Q, + Qb,

= moa +Mob ,

(6)

where (i) Toa, Tna, etc., satisfy stress equilibrium and force and moment equilibrium over the pipe cross-section, and (ii) 1"o8, Tnb, etc., are self-equilibrating thus producing no net forces or moments over the pipe cross-section, but which must be included to satisfy strain compatibility. When applied to a curved pipe, Novozhilov's stress equilibrium equations reduce to

--

(6TOa + 6 M o a ) + sinO (Tna + M n a ) q - ~ ( Zna + M na )

--

COS 0 sinp 0 Pa + ~0 ( t ~ P a ) + T O

~2 COS0 . 8T°a - ~ - ~ ( 6 M ° a ) + ---p"--- (T~1a --

6q~,

a - t ~ Q a =:

sin 0 3Mn~

Mna)

p

-3q~,

(7)

3Q a

O0

On = 6q*,

where p = R/r. Equilibrium is completely satisfied by these equations and they are consistent in that no forces or moments are generated by rigid body movements. The net forces and moments acting on a pipe cross-section if fig. 3 are given by Fx

- P a sin 0 + Qa cos 0

Fv

Tna

Fz =rEtfo2~ Pac°sO+QasinO Mx r(M~a + Tna) sin 0 Mv mz

dO.

(8)

rP~ - r( mna + Tna) cos 0

Thus, for given values of E~, Fy, F~, Mx, My, M~ acting on a pipe cross-section, expression for T~a, M,a, Pa, Qa are derived by eqs. (8); these expressions and values supplied for qo, G, q. are then used to give 8Toa, 6Mo~ by eqs. (7). It remains to determine the self-equilibrating components Tnb, M,b, Pb, Qb and the displacements, by

[L,]

u/," sin0

w/r u6 ~

(i)

~--

(ii)

sin 0 b O - v6 30

gO

6 30

12 302 6

v6

T.b

0

( t - v2)6

0 sin0 0 32 kv6-0 aO 802

3(6

(iii) (iv)

cos0

M~b

7632 I

6 + 12 302~ 302]

12(1 -

sin0

P

I '2)

3

v "~--~(6)

sinO ~(6) P cOzos__+v3 P

J.F. Whatham / Thin shell equationsfor circularpipe bends

81

[L2I

v/r sin 0 p

(i) (ii) (iii) (iv)

+

Qb

Pb

6(1 + ~)

3 3+'/

3+-/

~0

- - 6 3+y

O0

cos 0 P

-8

2(1 + v)(3 +V) O0 6

2 ( 1 + v ) ( 3 + 7 ) ~0 6

2(1 + v)(3 + y) ~0 ~

2(1 + v)(3 +-y)~0 (8 ~0)

3 +vy ~0(6) sinO p

3 ~_0(6) 3+-/

cos 0 P

the following equations derived in Appendix A. They are conveniently written, with matrix notation, as two groups of four equations:

I u/r

1 --~-~

-- (l -- p2)~T~a

- ~(1

- ,,:)~M~o

=

Pb

- ~ (~rOa - ,,~T~ + ~MOa - ~M~a)

Qb

~:ro~ - ,'~ r~a - OO: (SMOa - ,'~M~a)

02

v/r q~ [L~]

eb

Qb

6(1 + v)/JP

u/r 0 w/r - ~

M~

0 =

y 0 (~Pa) - ~Oa 3 + y O0

OMna +--ffC

.

(9)

--7 0 (6pa) +~JQa - - -OM~a 3 + - / O0

4. Specific problems For the solution of specific pipe problems, ignoring end effects, the six cases of 180 degree bends in fig. 4 have different end loadings but each is in equilibrium; it can be shown that the solution of four of these problems enables the effect of any force or moment loading on a pipe bend to be determined. For in-plane loading, case (a) and either one of cases (b) and (c) require solution, and for out-of-plane loading, case (d) and either of cases (e) and (f) must be solved. Cases (a), (b), (d) and (e) have been selected for further discussion. An example of superimposed solutions is shown at the bottom of fig. 4 where, for a 90 degree bend, shear force F acts on face A - A and axial force F in effect acts on face B-B. Non-linearity prevents the superimposing of pressure. The procedures for analyzing pipe bends under the following loading conditions are now considered:

82

J, F, Whatham / Thin shell equations for ~ircular pipe bendi!

(1) pressurized pipe bend, (2) in-plane bending moment, (3) in-plane bending moment with axial force, (4) out-of-plane shear force with twisting moment, and (5) twisting moment alone. The first step is to determine the pressures, forces and moments acting on the middle surface (table 1).

(o) i~ IN - PLANE

OUT - OF - PLANE

M

(b)

i

F~

(C)

F~

q

(b)

,'_r'-

{f) M¢ ~

(o)

B

/

Fig. 4. End loadings on 180 degree bends.

The next step is to find expressions for the equilibrium force and moment components 7~a, Mna, etc. by eqs. (8) and (7) (table 2). Finally, substituting in eq. (9) gives expressions for the self-equilibrating components Tnb, Mnb, Pb, Qb and the displacements in u, v, w, ~k (table 3) and allows their solution. The displacement terms involving X in cases (3) and (5) are required because the displacements in a 360 ° bend are not continuous, unlike the stresses and strains. The 8 dependence of these terms is derived from eqs. (A3) in Appendix A, their strain contribution being zero, and then the ~ dependence is obtained from the first, second, fifth and sixth equations of (9) by Fourier analysis using the total displacements. The truncation of each series in Appendix B to the minimum number of terms in the summation for convergence depended on the bend characteristic h (---Rt/r 2) for cases (1) and (2) and the ratioR/r for case (4). Six figure accuracy was achieved for: (i) the coefficient vl in cases (1) and (2) if h 2> (4.5/N) 33 for N up to 16, and (ii) the coefficient w1 in case (4) if R/r> (10/N) 1"2 for N up to 8.

J.F. Whathilm

/ Thin shell equations for circular pipe bends

83

Table I Reactions at cross-section X - X (refer to fig, 3) I N - P L A N E DEFLECTION

1

2 Mf F-~-~.

OUT-OF-PLANE DEFLECTION

4

3 M M~r~

5

M

M

-ii+ -cos01 Fx

I

-~- sinq~

! L

Fy

¢Tq~p

-M cos~

i

/

M sin q)

p acts on inside of pipe at radius q; qn acts on middle surface at radius r; ~ = 1 + ( r / R ) c o s 0. For equilibrium: pr, dO(R + r, cos 0) d~, = q, rdO(R + r c o s 0 ) d~.

Essentially, N equal to sixteen was adequate for cases (1) and (2) for h down to 0.015 and N equal to eight for case (4) for R/r down to 1.5. Substitution of the series expansions for the unknowns in eqs. (9) resulted in a number of unknown coefficients (bottom of table 3) and the solution required expanding the equations either by (i) Fourier analysis, or (ii) collocation. Since Fourier analysis ignored sin(N + 1)0 and cos(N + 1)19terms, the collocation method with 8 values the roots of sin(N + 1)6 = 0 or cos(N + 1)9 = 0, depending on the equation, gave the same result with less labour. However, the next step of adding exponentially decaying end effects by solving eqs. (9) with the right side zero (eigenvalue problem) could not be done conveniently except by Fourier analysis.

5. Pipe bend flexure by pressure or bending moment

The cross-sections of pipe bends (1) and (2) in table 3 remain plane after flexing, whereas the others do

J.F. Whatham / Thin shell equations for circular pipe bends

85

not, and the angle through which a cross-section at position ~/rotates in these cases is: /~ = VlO radians. Under pressure, bend (1) opens by vlr I~ = p * t

pT1 radians, E

P*tTl 1 r - fP 2 h

(10)

where

P* =priEr. Values calculated for the pressure factor fp, are given in table 5 for a range of h and show the close approximation to unity with Poisson's ratio p in the range 0.2 to 0.4. Under a pure bending moment, bend (2) cross-sections rotate by:

# :

v]

2M* .2M'7/=fb

¢3(1 -1'2) h

M~/ radians

(Beskin[7]).

(11)

" qrr2Et

Table 5 Pressure deflection factor fp (cf eq. (10))

Table 4 Straight pipe solutions

h 0.015 0.15 0.5 1 2

4 i

t/r

~'qr

0.01

0.1

0.3

1.000 1.000 1.000 1.000 1.000

0.996 1.003 1.000 0.998

1.011 1.001 0.980

u=0.2~0.4 _o r

121"12 - 24'£M%in O 12+ ~r

6(1"1))

"~

~M" Table 6 Bending deflection factor fb (cf. eq. (11))

121]2 - ~

M'cos e

M2b o (M2o :o)

12-~f . 1~ M cos e

T2b

12-Y ~ M

v

O (T2o= ri ) ] ] p ,

r .2

.

t/r

° cos u

24]] M ~ c o s 0 12"¥

0.01

24~ M ' c o s g 12.~r

Pb Qb

0

(P° = M ' )

0

(Qa=O)

Pri

U1

t

U2

2tr

pr i 2

M(r *z)cosO /Tr 3 t ( 1 + ~/i2 ) M(r*z) 21Tr 3t ( 1 + ~ 3 )

Correct S t ress Correction Fo¢ tor

membrane stress shown

1

M(r ÷z)cosO ] T r 3 t (1 ÷ ~/4 ] 1 ---~ 6

M(r*z) 2~'r3t (1+ ~/4) 1+

~1" 12

0.01 0.10 0.16 0.20 0.25 0.32 0.40 0.50 0.64 0.80 1.00 1.25 1.60 2.00 ~=0.2~0.4

0.999 1.001 0.996 0.982 0.964 0.944 0.937 0.952 1.007 1.100 1.243 1.447 1.755 2.124

0.1

0.3

0.992 0.990

0.954 0.956 0.970 1.018 1.106 1.244 1.443 1.747 2.112

86

J.F. Whatham / Thin shell equations jor circular pipe bend~

4°I

T

T

-'--P

T-T

. . . . . . .

]. . . . .

I

"

I

"

1

}

7

"i

F Y

. . . . . . . .

I

>I.-_] x bJ LL

I

1 QO1

01

1

2

BEND CHARACTE#ISTIC h

Fig. 5. Pipe bend flexibility in in-plane bending.

where

M* =-M/2~rr2Et. The ratio Vl/2M* denotes the flexibility factor f, since 2M*~/approximates the flexing from conventional beam theory, and is plotted in fig. 5 for a Poisson's ratio of 0.3. The bending factor fb was approximately unity for h < 0.3 (table 6 and linear portion of fig. 5), and this agrees with the Power Piping Standard [4] when Poisson's ratio is 0.3. For h > 0.3 the bending factors of thin walled tubes are given by the formulas of von Khrmhn [8] with h divided by v/1 - v 2 . Thus:

fb = A[ 105 +12408 A2 + 43200 A' ] 3 + 1608 A 2 + 43200 A 4

(12) '

which becomes for h > 0.7: fb = A [ ( 1 0 + 36 A 2 ) / ( 1 + 36A2)], where A = h/~/3(1

-v2),

f=fb/A.

Under out-of-plane twisting and shearing, analogous to a coil spring under a force F ( = cross-section of bend (4) in table 3 remains round but displaced out-of-plane by:

wlr . R(1 + p) ~iz = w, r i / - - 2M* "2M*ll =10 i ' + - ~

M/R), the

Mll

"rrrZEt "

The out-of-plane factor f0 approximates unity for Poisson's ratio in the range 0.2 to 0.4 (table 7).

(13)

J.F. Whatham / Thin shell equationsfor cir~lar pipe bends

87

Table 7 Out-of-plane displacement factor f0 (cf. eq. (12))

Table 8 Flexibility factors for in-plane bending

R/r

R/r

1.5 2 3 5 10

t/r 0,01

0.1

0.3

0,911 0.974 0.955 0.999 1.000

0.911 0.975 0.995 0.999 1.000

0.913 0.977 0.997 1.000 1.000

v:0.2~0,4

t/r

f Novozhilov

Fltigge

0.05 0.1

16.49 8.21

16.41 8.12

0.2 0.3

3.98 2.63

3.90 2.56

0.05 0.1

6,59 3.14

6.57 3.13

0.2 0.3

1.65 1.30

1,64 1.29

v=0.3, f=#Tr r2Et/M ~1

6. Wall thickness limitation Novozhilov limits the application of his theory to those shells having wall thickness-to-radius ratios t/R not greater than 0.05, but this evidently depends on the shell shape and the manner of loading. Shear strain through the pipe wall, which is neglected by assumption (iii) in section 1, becomes more significant as the wall thickness is increased. However, as far as the pure bending moment case (2) is concerned, the shear stress and hence the shear strain through the wall is small and thicker walls are permitted. Three tests of the theory were made. Firstly, solutions for straight pipes in table 4 were obtained for cases (1), (2) and (4); these stresses closely approximated those form conventional strength of materials theory, the correction factors being 1 - 3'/6 for case (2) and 1 + 3,/12 for case (3) giving differences of 1.5% and 0.75% respectively for t/r-0.3. Secondly, the flexibility factor with a constant bend characteristic changed less than 3% as t,/R was increased from 0.01 to 0.3, table 6 demonstrates this with Poisson's ratio ranging from 0.2 to 0.4. Thirdly, flexibility factors calculated by Thompson [5] by the Fliigge thicker shell theory (Kraus [6]) agreed within 3% with those from the Novozhilov theory for t/r ratios up to 0.3 (table 8).

7. Conclusions Equations have been presented for the analysis of pipe bends under pressure or any end loading, making use of superposltion and neglecting end effects although non-linearity prevents the superposition of pressurizing on end loading solutions, and the theory is effective for pipe bends with t/r ratios up to 0.3. The deflections produced by pressurizing, by pure in-plane bending and by out-of-plane bending under constant shear and torsion (as in a coil spring) have been calculated for a range of v, R/r and t/r, and factors which in most cases are unity, have been derived for use with general formulas. The manner in which end effects from, the example, stiff flanges or straight tangents can be added to give the results reported by Whatham and Thompson [3] will be the subject of a later publication.

8. Acknowledgement The author acknowledges the advice and encouragement of Professor J.J. Thompson of the School of Nuclear Engineering, University of New South Wales.

J.F. Whatham / Thin shell equationsfor circularptpe bends

88

Appendix A. Derivation of system equations Referring to fig. 2 and using Novozhilov's notation, the pipe bend parameters are: Ot1 = 0 ,

0/2 z ~,

R 1 = r,

R2

_Rc o s 0

ds 1 A I --d0/1 - r ,

A2

--- d0/2 - 6 r ,

t-r=

co@0r ,

(A.1)

ds2

where cos 0 6=1+--, P

p = R/r,

~1= pep.

Substituting the pipe bend parameters into Novozhilov's equations for the resultants shown as P and Q in fig. 3 we get P = S + 2H/r, 1 [3Mn

3,~ +

a=;~t

_36H]

2--~-].

(A.2)

The strain distributions in the pipe wall are given by: transverse strain

co(z) = % + z x o,

longitudinal strain

%( z ) = % + z xn,

shear strain

w(z)=w+z

(oo) 2~-

R,

R-2

in which the strain parameters are related to the displacements by Novozhilov's equations written in terms of the pipe bend parameters. Thus, 1[3u

]

1 [3v

'0:r[~+w'

'~:~r

sin0

37

p

cos0

U+--W,

]

p

l[3u+sL.0..... v ] q 13v

w=6-r [ 3~

p

.o = - ~

~o2 '

--~

ao

r D0'

]

(A.3) ~ =~

~ + or

~-

u

,

1 [35_~+sin0 ] 13¢

¢ - 6r z

p

v

-}-----

r DO

where 1

cos0 v--~-~ •

The usual force-strain relations of a linear shell theory apply, namely: E1 To - 1 ---1,2 ( % + v% ),

E t (on + w o ) , T, -- 1 -- v 2

M° = 12(1Et3 - vz)

M. - 12(1 - .2)

Et

S=2(l+v)

(x. + ~ . ) ,

Et3

Et 3

w,

H-12(1+v)

""

(Kn + w o ) ,

(A.4)

J.F. Whatham / Thin shell equations for circularpipe bends

89

The object is now to express the derivatives with respect to ~ of specified unknowns Mn, Tn, P, Q,/t, w, v and + in terms of derivatives with respect to 0 and to separate the resultants into their two components (cf. eq. (6)). The required eight eqs. (9) are derived from: (i) Eqs. (A.2) for aM,~/O71. (ii) Eqs. (7) with subscripts 'a' replaced by 'b' and the q terms zero. (iii) The second, third, fifth and seventh equations of (A.3), making use of eqs. (A.4).

Appendix B. Series expansions for the unknowns In-plane

Out-of-plane N

~=ulsin0+

N

~ u. sinnO

fi=-ulcos0-

~ uncosnO

n=2

n=2 N

N

~ = Wo + Wt cos O + ]~ w. cos nO

~=wlsin0+

n=2

~ w. sin nO n=2

N

]~ M. cosn0

M~=M o+M,cosO+

N

M,=M,

sinO+ E M. sinn0

n=2

n=2 N

N

T. = - M, cos 0 - E T. cos nO

frn = - M, sin 0 -

n=2

E T,, sin nO n=2

N

N

~-I) 1COS0~- E v. cosnO

~ = v 1 s i n 0 + ~ v. sinnO

n--2

n=2 N

N

q~=~ko +~p, c o s 0 + ~] q,ncosn0

q~=~, sin0 ~ qJ. sin n0

n=2

n=2

N

N

P = Q , s i n 0 + ~ P. sinn0

l f i = - Q , c o s O - ~] P~cosn0

n=2

n=2 N

N

Q = Q o + Q , cosO+ ~ Q. cosnO n=2

Q=Qlsin0+

]~ Q. sinnO n=2

Note: (i) Coefficients are different for in-plane and out-of-plane unknowns. (ii) Coefficients v o and u 0 have been omitted since they represent indeterminate rigid body movements in cases (3) and (5) respectively.

References [1] [2] [3] [4] [5]

V.V. Novozhilov, Thin Shell Theory (Wolters-Noordhoff, Groningen, 1970). J.F. Whatharn, Trans. Inst. Eng. (Aust.) CE 21(2) (1979) 80. J.F. Whatham and J.J. Thompson, Nucl. Engrg. Des. 54 (1979) 17. Power Piping Standard ANSI-B 31.1 (ASME, New York, 1977). J.J. Thompson, Shell theory analysis of pure in-plane bending of a pipe bend, Trans. Third Int. Conf. Struct. Mech. in Reactor Technology, London, 1-5 Sept., 1975, Paper F6/1. [6] H. Kraus, Thin Elastic Shells (Wiley, New York, 1967). [7] L. Beskin, Bending of curved thin tubes, Appl. Mech. Trans. ASME 12 (1945) A - 1. [8] Th. von Kbxmhn, Zeitschrift des VDI 55 (1911) 1889.